ITU-T E 507-1993 MODELS FOR FORECASTING INTERNATIONAL TRAFFIC《国际话务预测模型》.pdf

1、INTERNATIONAL TELECOMMUNICATION UNION)45G134 % TELECOMMUNICATIONSTANDARDIZATION SECTOROF ITU4%,%0(/.%G0G0.%47/2+G0G0!.$G0G0)3$.15!,)49G0G0/k, k = 1, . p are the autoregressive parameters.The model is denoted by AR(p) since the order of the model is p.By use of regression analysis the estimates of th

2、e parameters can be found. Because of common trends theexogenous variables (Xt1, Xt2, . Xtp) are usually strongly correlated. Hence the parameter estimates will becorrelated. Furthermore, significance tests of the estimates are somewhat difficult to perform.Another possibility is to compute the empi

3、rical autocorrelation coefficients and then use the Yule-Walkerequations to estimate the parameters k. This procedure can be performed when the time series Xt are stationary.If, on the other hand, the time series are non stationary, the series can often be transformed to stationarity e.g., bydiffere

4、ncing the series. The estimation procedure is given in Annex A, A.1.Fascicle II.3 - Rec. E.507 53.4 Autoregressive integrated moving average (ARIMA) modelsAn extention of the class of autoregressive models which include the moving average models is calledautoregressive moving average models (ARIMA m

5、odels). A moving average model of order q is given by:Xa a a att t t qtq= 11 2 2. (3-8)whereatis white noise at time t; k are the moving average parameters.Assuming that the white noise term in the autoregressive models in 3.3 is described by a moving averagemodel, one obtains the so-called ARIMA (p

6、, q) model:XX X Xaa a att t ptptt tqtq=+ + 11 2 2 1122. . (3-9)The ARIMA model describes a stationary time series. If the time series is non-stationary, it is necessary todifference the series. This is done as follow:Let Ytbe the time series and B the backwards shift operator, thenXBYtdt=()1 (3-10)w

7、hered is the number of differences to have stationarity.The new model ARIMA (p, d, q) is found by inserting equation (3-10) into equation (3-9).The method for analyzing such time series was developed by G. E. P. Box and G. M. Jenkins 3. To analyzeand forecast such time series it is usually necessary

8、 to use a time series program package.As indicated in Figure 1/E.507 a tentative model is identified. This is carried out by determination of necessarytransformations and number of autoregressive and moving average parameters. The identification is based on thestructure of the autocorrelations and p

9、artial autocorrelations.The next step as indicated in Figure 1/E.507 is the estimation procedure. The maximum likelihood estimates areused. Unfortunately, it is difficult to find these estimates because of the necessity to solve a nonlinear system ofequations. For practical purposes, a computer prog

10、ram is necessary for these calculations. The forecasting model isbased on equation (3-9) and the process of making forecasts l time units ahead is shown in A.2.The forecasting models described so far are univariate forecasting models. It is also possible to introduceexplanatory variables. In this ca

11、se the system will be described by a transfer function model. The methods foranalyzing the time series in a transfer function model are rather similar to the methods described above.Detailed descriptions of ARIMA models are given in 1, 2, 3, 5, 11, 15 and 17.3.5 State space models with Kalman Filter

12、ingState space models are a way to represent discrete-time process by means of difference equations. The statespace modelling approach allows the conversion of any general linear model into a form suitable for recursiveestimation and forecasting. A more detailed description of ARIMA state space mode

13、ls can be found in 1.For a stochastic process such a representation may be of the following form:XXZtttt+=+1 (3-11)andYHXvttt=+ (3-12)6 Fascicle II.3 - Rec. E.507whereXtis an s-vector of state variables in period t,Ztis an s-vector of deterministic events, is an s s transition matrix that may, in ge

14、neral, depend on t,tis an s-vector of random modelling errors,Ytis a d-vector of measurements in period t,H is a d s matrix called the observation matrix, andvtis a d-vector of measurement errors.Both tin equation (3-11) and vtin equation (3- 12) are additive random sequences with known statistics.

15、Theexpected value of each sequence is the zero vector and tand vtsatisfy the conditions:EQtjTttj = for all t, j,(3-13)Evv RtjTttj= for all t, jwhereQtand Rtare nonnegative definite matrices,2)andtjis the Kronecker delta.Qtis the covariance matrix of the modelling errors and Rtis the covariance matri

16、x of the measurement errors;the tand the vtare assumed to be uncorrelated and are referred to as white noise. In other words:EvtjT=0for all t, j, (3-14)andEvXtT00= for all t. (3-15)Under the assumptions formulated above, determine Xt,tsuch that:EX X X Xtt tTtt t()(),=minimum, (3-16)whereXt,tis an es

17、timate of the state vector at time t, andXtis the vector of true state variables._2)A matrix A is nonnegative definite, if and only if, for all vectors z, zTAz 0.Fascicle II.3 - Rec. E.507 7The Kalman Filtering technique allows the estimation of state variables recursively for on-line applications.T

18、his is done in the following manner. Assuming that there is no explanatory variable Zt, once a new data point becomesavailable it is used to update the model:XX KYHXtt tt t t tt, ,()=+11(3-17)whereKtis the Kalman Gain matrix that can be computed recursively 18.Intuitively, the gain matrix determines

19、 how much relative weight will be given to the last observed forecasterror to correct it. To create a k-step ahead projection the following formula is used:XXtktktt+=, (3-18)whereXt+k,tis an estimate of Xt+kgiven observations Y1, Y2, ., Yt.Equations (3-17) and (3-18) show that the Kalman Filtering t

20、echnique leads to a convenient forecastingprocedure that is recursive in nature and provides an unbiased, minimum variance estimate of the discrete time processof interest.For further studies see 4, 5, 16, 18, 19 and 22.The Kalman Filtering works well when the data under examination are seasonal. Th

21、e seasonal traffic load datacan be represented by a periodic time series. In this way, a seasonal Kalman Filter can be obtained by superimposing alinear growth model with a seasonal model. For further discussion of seasonal Kalman Filter techniques see 6 and20.3.6 Regression modelsThe equations (3-1

22、) and (3-2) are typical regression models. In the equations the traffic, Yt, is the dependent (orexplanatory) variable, while time t is the independent variable.A regression model describes a linear relation between the dependent and the independent variables. Givencertain assumptions ordinary least

23、 squares (OLS) can be used to estimate the parameters.A model with several independent variables is called a multiple regression model. The model is given by:YXX Xutttkkt t=+ + + + 011 22.(3-19)whereYtis the traffic at time t,i , i = 0, 1, ., k are the parameters,Xit , ie= 1, 2, ., k is the value of

24、 the independent variables at time t,utis the error term at time t.Independent or explanatory variables which can be used in the regression model are, for instance, tariffs,exports, imports, degree of automation. Other explanatory variables are given in 2 “Base data for forecasting“ inRecommendation

25、 E.506.Detailed descriptions of regression models are given in 1, 5, 7, 15 and 23.8 Fascicle II.3 - Rec. E.5073.7 Econometric modelsEconometric models involve equations which relate a variable which we wish to forecast (the dependent orendogenous variable) to a number of socio-economic variables (ca

26、lled independent or explanatory variables). The formof the equations should reflect an expected casual relationship between the variables. Given an assumed model form,historical or cross sectional data are used to estimate coefficients in the equation. Assuming the model remains validover time, esti

27、mates of future values of the independent variables can be used to give forecasts of the variables ofinterest. An example of a typical econometric model is given in Annex C.There is a wide spectrum of possible models and a number of methods of estimating the coefficients (e.g., leastsquares, varying

28、 parameter methods, nonlinear regression, etc.). In many respects the family of econometric modelsavailable is far more flexible than other models. For example, lagged effects can be incorporated, observationsweighted, ARIMA residual models subsumed, information from separate sections pooled and par

29、ameters allowed tovary in econometric models, to mention a few.One of the major benefits of building an econometric model to be used in forecasting is that the structure or theprocess that generates the data must be properly identified and appropriate causal paths must be determined. Explicitstructu

30、re identification makes the source of errors in the forecast easier to identify in econometric models than in othertypes of models.Changes in structures can be detected through the use of econometric models and outliers in the historical dataare easily eliminated or their influence properly weighted

31、. Also, changes in the factors affecting the variables inquestion can easily be incorporated in the forecast generated from an econometric model.Often, fairly reliable econometric models may be constructed with less observations than that required for timeseries models. In the case of pooled regress

32、ion models, just a few observations for several cross-sections are sufficientto support a model used for predictions.However, care must be taken in estimating the model to satisfy the underlying assumptions of the techniqueswhich are described in many of the reference works listed at the end of this

33、 Recommendation. For example the numberof independent variables which can be used is limited by the amount of data available to estimate the model. Also,independent variables which are correlated to one another should be avoided. Sometimes correlation between thevariables can be avoided by using dif

34、ferenced or detrended data or by transformation of the variables. For furtherstudies see 8, 12, 13, 14 and 21.4 Discontinuities in traffic growth4.1 Examples of discontinuitiesIt may be difficult to assess in advance the magnitude of a discontinuity. Often the influence of the factorswhich cause dis

35、continuties is spread over a transitional period, and the discontinuity is not so obvious. Furthermore,discontinuities arising, for example, from the introduction of international subscriber dialling are difficult to identifyaccurately, because changes in the method of working are usually associated

36、 with other changes (e.g. tariff reductions).An illustration of the bearing of discontinuities on traffic growth can be observed in the graph of Figure4/E.507.Discontinuities representing the doubling and even more of traffic flow are known. It may also be notedthat changes could occur in the growth

37、 trend after discontinuities.In short-term forecasts it may be desirable to use the trend of the traffic between discontinuities, but for long-term forecasts it may be desirable to use a trend estimate which is based on long-term observations, including previousdiscontinuities.In addition to random

38、fluctuations due to unpredictable traffic surges, faults, etc., traffic measurements are alsosubject to systematic fluctuations, due to daily or weekly traffic flow cycles, influence of time differences, etc.4.2 Introduction of explanatory variablesIdentification of explanatory variables for an econ

39、ometric model is probably the most difficult aspect ofeconometric model building. The explanatory variables used in an econometric model identify the main sources ofinfluence on the variable one is concerned with. A list of explanatory variables is given in Recommendation E.506, 2.Fascicle II.3 - Re

40、c. E.507 9Economic theory is the starting point for variable selection. More specifically, demand theory provides thebasic framework for building the general model. However, the description of the structure or the process generating thedata often dictate what variables enter the set of explanatory v

41、ariables. For instance, technological relationship mayneed to be incorporated in the model in order to appropriately define the structure.Although there are some criteria used in selecting explanatory variables e.g., R2, Durbin-Watson (D-W)statistic, root mean square error (RMSE), ex-post forecast p

42、erformance, explained in the references, statisticalproblems and/or availability of data (either historical or forecasted) limit the set of potential explanatory variables andone often has to revert to proxy variables. Unlike pure statistical models, econometric models admit explanatoryvariables, no

43、t on the basis of statistical criteria alone but, also, on the premise that causality is, indeed, present.A completely specified econometric model will capture turning points. Discontinuities in the dependentvariable will not be present unless the parameters of the model change drastically in a very

44、 short time period.Discontinuities in the growth of telephone traffic are indications that the underlying market or technological structurehave undergone large changes.Sustained changes in the growth of telephone demand can either be captured through varying parameterregression or through the introd

45、uction of a variable that appears to explain the discontinuity (e.g., the introduction of anadvertising variable if advertising is judged to be the cause of the structural change). Once-and-for-all, or step-wisediscontinuities, cannot be handled by the introduction of explanatory changes: dummy vari

46、ables can resolve thisproblem.4.3 Introduction of dummy variablesIn econometric models, qualitative variables are often relevant; to measure the impact of qualitative variables,dummy variables are used. The dummy variable technique uses the value 1 for the presence of the qualitative attributethat h

47、as an impact on the dependent variable and 0 for the absence of the given attribute.Thus, dummy variables are appropriate to use in the case where a discontinuity in the dependent variable hastaken place. A dummy variable, for example, would take the value of zero during the historical period when c

48、alls wereoperator handled and one for the period for which direct dial service is available.Dummy variables are often used to capture seasonal effects in the dependent variable or when one needs toeliminate the effect of an outlier on the parameters of a model, such as a large jump in telephone dema

49、nd due to apostal strike or a sharp decline due to facility outages associated with severe weather conditions.Indiscriminate use of dummy variables should be discouraged for two reasons:1) dummy variables tend to absorb all the explanatory power during discontinuities, and10 Fascicle II.3 - Rec. E.5072) they result in a reduction in the degrees of freedom.5 Assessing model specification5.1 GeneralIn this section metho